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Semilinear Schrödinger Models

In: Methods for Partial Differential Equations

Author

Listed:
  • Marcelo R. Ebert

    (University of São Paulo, Department of Computing and Mathematics)

  • Michael Reissig

    (TU Bergakademie Freiberg, Institute of Applied Analysis)

Abstract

In this chapter we introduce results for semilinear Schrödinger models with power nonlinearity in the focusing and defocusing cases as well. First of all, we show how by a scaling argument a proposal for a critical exponent appears. This critical exponent heavily depends on the regularity of the data. The issue of L 2 and H 1 data is explained. As for the linear Schrödinger equation (see Sect. 11.2.3 ), some conserved quantities are given. Then, a global (in time) well-posedness result is proved for weak solutions in the subcritical L 2 case. This result is valid for both cases focusing and defocusing, respectively. Finally, the subcritical H 1 case is treated. Here the main concern is to show differences between both focusing and defocusing cases. A local (in time) well-posedness result is proved. This result contains, moreover, a blow up result in the focusing and a global (in time) well-posedness result in the defocusing case.

Suggested Citation

  • Marcelo R. Ebert & Michael Reissig, 2018. "Semilinear Schrödinger Models," Springer Books, in: Methods for Partial Differential Equations, chapter 0, pages 367-382, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66456-9_21
    DOI: 10.1007/978-3-319-66456-9_21
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